Do protests actually work?

You need to define your goals. Do protests work to achieve what?

In my area, protests are largely social gatherings of like minded people. I asked protesters on three occasions last year and only two of a couple dozen protesters thought they were reaching an audience that does not already agree. I stress this was not a scientific study, but at least average for anecdote.

According to Stephen Pinker, protests can turn individual knowledge into mutual knowledge

In individual knowledge, A knows X and B knows X. In mutual knowledge A knows X, B knows X, A knows that B knows X; B knows and A knows X; A knows that B knows that A knows X ad infinitum. And this is a difference that has profound consequences.

For example, why is freedom of assembly enshrined as a fundamental right in a democracy and why are political revolutions often triggered when a crowd gathers in a public square to challenge the president in his palace. Well it is because

when people were at home everyone knew that they loathed the dictator, but no-one knew that other people knew that other people knew that they knew. Once you assemble in a place where everyone can see everyone else everyone knows that everyone else knows that everyone else knows that the dictator is loathed, and that gives them the collective power to challenge the authority of the dictator who otherwise could kick off dissenters one at a time.

This suggests that protests may lead to something in a particular case when most people already have individual knowledge, but they do not have mutual knowledge yet. For example, suppose those people really care about some issue and have idea what to do, then if participating in protests is risky, that signals that all those protesters are willing to take risks in order to achieve their goal (curiously, in this particular case, if protesting is safe (as it is in most Western countries), the signal might be less clear). This way individual knowledge becomes mutual knowledge. So if it is the lack of mutual knowledge that prevents their goals from being achieved, then protests might help. Otherwise, if it is something else that prevents solution (e.g.lack of idea how to solve a problem, various game theoretic (or coordination) problems that are not solved by going from individual to mutual knowledge, etc.) from being achieved, they are probably much more likely to be useless.

Are e.g. the Ferguson/police crime protests a good way of attacking the problem?

What problem? That Blacks aren't free to steal from and intimidate Asian store owners and then charge at a police officer going for his gun?

I was wondering that too; personally, I have no idea how to even begin answering the question. It would seem that at least some protests do work, as evidenced by the civil rights movement during the Martin Luther King era; but I don't know if this is true in general.

Have a look at this post from Death is Bad Blog. It won't answer your questions, but it will help you shine more light on it.